It's not harder to run than any other game, I suppose. The mechanics are often the easy part in GMing anyway. I've started GMing DFRPG with only a little bit of GM experience in other games and zero experience in the DFRPG and it turned out pretty well, I would say. It is definitely doable. Your group should know that you are doing this for the first time, so they should show a bit of patience and maybe help out as best they can. Playing the game is a group effort as much as it is work for the GM.
When it comes to improv, there's improv and then there's improv. Savvy?
Ok, let me explain. You can just go into a game mostly blind and just let things happen as they may. This is pure improv, and it's something that I think only very few people can truly do. I certainly can't.
The other kind of improv is easier, but it requires a lot of prep work. Instead of planning a story or something along those lines, you plan an opposition. You map out who the bad guys are, what they can do, what they have at their disposal, what they need to do to reach their goal, what their goal is, and so on. Plan ahead to what would happen if nothing goes wrong, because the bad guy will most certainly have that plan.
Once you've done that, you can have the players happen upon the bad guys at some point. That will most likely hinder, at least inconvenience the bad guys. Maybe one of the player characters is even a target or plays a vital role in the plan, so getting them involved should be easy. Now comes the trick: because you know what the bad guys need and what they can do to get it, you can now adjust their plans to the players actions. If they foil one end of the plan, you can adjust easily and go forth with another part. Improvising this is far easier, since you don't really improvise, you just do what logically follows. That's a big burden off the whole improv pressure.
Of course it can still happen that the players do something you are completely overwhelmed by. When that happens, don't panic (he said in bright, friendly letters). You can always just say "wow, guys, you caught me totally off guard, give me a few minutes to think about this". That takes the pressure away from coming up with something RIGHT NOW, and the rest of the group can take the time to get a smoke, go to the bathroom, clean up a bit or look something up.
But you can also involve the players in planning the scene. Tell them this is something you didn't expect, so you don't really know what to do here, but they can contribute some ideas what should be in a scene like this. It might be a bit weird to all of you at first, because that's not something an RPG usually does, but it works quite well in Fate.
There are a few ready made case files available on drivethrurpg. You could look into those to run as your first adventures. They are pretty well structured and do a lot of the heavy lifting for you. Once you've gone through those, you'll have a bit of experience in running the game and should be good to go your own way. The case files are available for free:
http://www.drivethrurpg.com/product/87671/Dresden-Files-RPG-Casefile-Neutral-Groundshttp://www.drivethrurpg.com/product/90998/Dresden-Files-RPG-Casefile-Night-Fearshttp://www.drivethrurpg.com/product/94745/Dresden-Files-RPG-Casefile-Evil-Acts